New Journal of Physics

Transcription

1 New Journal of Phsics The open access journal for phsics Negative phase velocit in a material with simultaneous mirror-conjugated and racemic chiralit characteristics Tom G Macka 1,3 and Akhlesh Lakhtakia 2 1 School of Mathematics, Universit of Edinburgh, James Clerk Mawell Building, The King s Buildings, Edinburgh EH9 3JZ, UK 2 CATMAS Computational and Theoretical Materials Sciences Group, Department of Engineering Science and Mechanics, Pennslvania State Universit, Universit Park, PA , USA T.Macka@ed.ac.uk and al4@psu.edu New Journal of Phsics 7 (2005) 165 Received 22 March 2005 Published 8 August 2005 Online at doi:1088/ /7/1/165 Abstract. The propagation of electromagnetic plane waves in a material with simultaneous mirror-conjugated and racemic chiralit (SMCRC) characteristics is investigated. General conditions for negative phase velocit (NPV) (i.e., phase velocit directed opposite to the time-averaged Ponting vector) are derived for both unirefringent and birefringent propagation. Through numerical studies, it is demonstrated that NPV propagation arises provided that the magnitude of the magnetoelectric constitutive parameter is sufficientl large compared with the magnitudes of dielectric and magnetic constitutive parameters. However, the relative magnitude of the magnetoelectric constitutive parameter has little bearing upon the directions, which support NPV propagation. The propensit for NPV propagation is much enhanced through incorporating dielectric and magnetic constitutive parameters, which are negative-real. A wide range of constitutive parameter values are considered in order to accommodate the possibilities offered b the fabrication of artificial SMCRC materials. 3 Author to whom an correspondence should be addressed. New Journal of Phsics 7 (2005) 165 PII: S (05) /05/ $30.00 IOP Publishing Ltd and Deutsche Phsikalische Gesellschaft

2 2 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Contents 1. Introduction 2 2. Analsis Preliminaries General propagation Unirefringent propagation along the racemic ais Birefringent propagation Numerical results Concluding remarks 14 Acknowledgment 15 References Introduction The phase velocit of an electromagnetic plane wave is described as negative, if it is directed opposite to its time-averaged Ponting vector. Several interesting consequences follow as a result of negative phase velocit (NPV) propagation [1]. Most prominent amongst these is the phenomenon of negative refraction. Eperimental reports of microwave negative refraction in artificial metamaterials [2] [4] have provoked considerable ecitement in the electromagnetics and materials communities within the past few ears. Recent efforts have been directed towards achieving negative refraction at higher frequencies, with the ultimate goal being optical negative refraction [5]. In view of the practical difficulties involved in achieving negative refraction in isotropic dielectric-magnetic materials, attention latel has been directed towards more comple materials, such as anisotropic [6, 7] and bianisotropic [8] materials as well as isotropic chiral materials [9] [11]. We report here on the prospects for NPV propagation in a particular class of anisotropic chiral materials. The simplest manifestation of chiralit arises in isotropic chiral materials (ICMs) [12]. These are specified b frequenc-domain constitutive relations of the form D = ɛe ± ξh and B = ξe + µh, (1) where the material described b the lower signs is the mirror conjugate of the material described b the upper signs. The homogeneous miture in equal proportions of an ICM and its mirror conjugate, as specified b (1), is called racemic. The racemic miture is an isotropic dielectricmagnetic medium specified b the constitutive relations: D = ɛe and B = µh, (2) and is therefore similar to the materials commonl investigated for negative refraction [13]. However, it is possible to combine in equal proportions an ICM and its mirror-conjugate in a homogeneous miture and et retain optical activit. Such a miture must be effectivel an anisotropic continuum and can be understood as follows. Suppose an artificial ICM is fabricated b randoml dispersing and randoml orienting miniature left-handed springs in some host material [12, 14]. Its mirror-conjugate can be fabricated b similarl dispersing miniature springs New Journal of Phsics 7 (2005) 165 (

3 3 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT in an identical host material, with the eception that the springs must be right handed. In a racemic material, left- and right-handed springs must be dispersed in equal proportions [15]. Anisotropic chiralit can be induced if all springs in an artificial material are of the same handedness and are aligned parallel to each other, as has been demonstrated satisfactoril [16, 17]. Even more complicated anisotropic chiral behaviour can result if left-handed springs are aligned along a fied ais, and right-handed springs along some other fied ais. Our focus here is on materials which displa simultaneous mirror-conjugated and racemic chiralit (SMCRC) characteristics. Artificial SMCRC materials can be conceived as containing left-handed springs aligned along the -ais, right-handed springs along the -ais, and leftas well as right-handed springs in equal proportions aligned along the -ais [18]. Other schemes for artificial fabrication also have been offered [19]. We emphasie that the prospect of fabricating artificial SMCRC materials allows a wide range of constitutive parameter values to be contemplated. Ver significantl, SMCRC materials occur in nature in the form of certain minerals of the tpe RXO 4, where R is a trivalent rare earth, X is either vanadium or arsenic or phosphorus and O is ogen [20, 21]. The SMCRC characteristics are displaed below the Néel temperatures. Details pertaining to the general propagation properties in SMCRC materials are available elsewhere [18, 19]. In the following sections, we derive conditions for NPV propagation in an SMCRC material. Both unirefringent and birefringent propagation are considered for generall dissipative materials. The analtic epressions developed are eplored numericall b means of representative eamples. In the notation adopted, vectors are underlined, whereas dadics are double underlined. Unit vectors are denoted b the ˆ smbol, while I =ˆ ˆ + ŷ ŷ + ẑ ẑ is the identit dadic. The comple conjugate of a comple-valued quantit q is represented b q, whereas its real and imaginar parts are written as Re{q} and Im{q}, respectivel. All electromagnetic field phasors and constitutive parameters depend implicitl on the circular frequenc ω of the electromagnetic field and an ep( iωt) time-dependence is implicit. The free-space (i.e., vacuum) wavenumber is denoted b k 0 = ω ɛ 0 µ 0 with ɛ 0 and µ 0 being the permittivit and permeabilit of free space, respectivel. 2. Analsis 2.1. Preliminaries The propagation of plane waves with field phasors E(r) = E 0 ep(ikû r) and H(r) = H 0 ep(ikû r) (3) is considered in an SMCRC material, whose frequenc-domain constitutive relations [18] D(r) = ɛ E(r) + ξ H(r) and B(r) = ξ E(r) + µ H(r) (4) emplo the constitutive dadics ɛ = ɛ 1 (ˆ ˆ + ŷ ŷ) + ɛ 3 ẑ ẑ, ξ = ξ(ˆ ˆ ŷ ŷ), µ = µ 1 (ˆ ˆ + ŷ ŷ) + µ 3 ẑ ẑ. (5) New Journal of Phsics 7 (2005) 165 (

4 4 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Eamples of natural materials described b (4) are GdVO 4 and HoPO 4 [20, 21]. Generall, the constitutive parameters are comple-valued, but if dissipation is neglected, we have ɛ 1,3 R, µ 1,3 R and iξ R. The wavenumber k in (3) is also comple-valued in general; i.e., k = k R +ik I (k R,k I R). (6) It ma be calculated from the planewave dispersion relation wherein det[l(ikû)] = 0, (7) L(a) = (a I +iωξ) µ 1 (a I +iωξ) ω 2 ɛ. (8) Of particular interest is the orientation of the phase velocit, as specified b the direction of k R û, relative to the direction of power flow given b the time-averaged Ponting vector P(r) t = (1/2) Re[E(r) H (r)]. The combination of the constitutive relations (4) with the source-free Mawell curl postulates ields P(r) t = 1 2 ep( 2k Iû r)re { E 0 [ (µ 1 ) ( k ω û E 0 + ξ E 0 for plane waves (3). B definition, NPV propagation occurs when )]} k R û P(r) t < 0. (10) (9) 2.2. General propagation For planewave propagation in an arbitrar direction, i.e., û = (sin θ cos φ, sin θ sin φ, cos θ) (0 θ π; 0 φ 2π), (11) the [L] lm components of L(ik û) are given as [L] 11 = k2 sin 2 θ sin 2 φ µ 3 + k2 cos 2 θ ω 2 (ɛ 1 µ 1 + ξ 2 ) µ 1, (12) [L] 12 = k2 sin 2 θ sin φ cos φ µ 3, (13) k sin θ(k cos θ cos φ + ωξ sin φ) [L] 13 =, (14) µ 1 [L] 21 = [L] 12, (15) New Journal of Phsics 7 (2005) 165 (

5 5 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT [L] 22 = k2 sin 2 θ cos 2 φ + k2 cos 2 θ ω 2 (ɛ 1 µ 1 + ξ 2 ), µ 3 µ 1 (16) k sin θ(k cos θ sin φ + ωξ cos φ) [L] 23 =, µ 1 (17) k sin θ(k cos θ cos φ ωξ sin φ) [L] 31 =, µ 1 (18) k sin θ(k cos θ sin φ ωξ cos φ) [L] 32 =, µ 1 (19) [L] 33 = k2 sin 2 θ ω 2 ɛ 3, (20) µ 1 and the dispersion relation (7) reduces to the k 2 -quadratic polnomial k 4 [(ɛ 1 sin 2 θ + ɛ 3 cos 2 θ)(µ 1 sin 2 θ + µ 3 cos 2 θ) + ξ 2 sin 4 θ cos 2 2φ] k 2 ω 2 (ɛ 1 µ 1 + ξ 2 ) [(ɛ 3 µ 1 + ɛ 1 µ 3 ) sin 2 θ +2ɛ 3 µ 3 cos 2 θ]+ɛ 3 µ 3 ω 4 (ɛ 1 µ 1 + ξ 2 ) 2 = 0. (21) The k-roots of (21) provide the four wavenumbers k = k 1, k 2, k 3 and k 4 as (ɛ 1 µ 1 + ξ k 1 = ω 2 )[2ɛ 3 µ 3 cos 2 θ + sin 2 θ(ɛ 3 µ 1 + ɛ 1 µ 3 (ɛ 3 µ 1 ɛ 1 µ 3 ) 2 4ɛ 3 µ 3 ξ 2 cos 2 2φ)], 2[(ɛ 1 sin 2 θ + ɛ 3 cos 2 θ)(µ 1 sin 2 θ + µ 3 cos 2 θ) + ξ 2 sin 4 θ cos 2 2φ] (22) k 2 = k 1, (23) (ɛ 1 µ 1 + ξ k 3 = ω 2 )[2ɛ 3 µ 3 cos 2 θ + sin 2 θ(ɛ 3 µ 1 + ɛ 1 µ 3 + (ɛ 3 µ 1 ɛ 1 µ 3 ) 2 4ɛ 3 µ 3 ξ 2 cos 2 2φ)], 2[(ɛ 1 sin 2 θ + ɛ 3 cos 2 θ)(µ 1 sin 2 θ + µ 3 cos 2 θ) + ξ 2 sin 4 θ cos 2 2φ] (24) k 4 = k 3. In consideration of the rate of energ flow, as provided b the time-averaged Ponting vector (9), we remark that onl two of the four k-roots correspond to propagating plane waves, for each propagation direction {θ, φ}. We denote the wavenumbers of the propagating modes b k a and k b where { k1 if Im{k 1 } 0, k a = (26) k 2 if Im{k 2 } > 0, { k3 if Im{k 3 } 0, k b = (27) k 4 if Im{k 4 } > 0. Thus, the SMCRC material is generall birefringent. However, there are eceptions: the chosen material is unirefringent with respect to propagation along the racemic (i.e., ) ais. That is, for θ = 0, onl one independent wavenumber emerges from the dispersion relation (21). New Journal of Phsics 7 (2005) 165 ( (25)

6 6 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Furthermore, pathological unirefringence [22] arises when ɛ 3 µ 1 ɛ 1 µ 3 = 2 ɛ 3 µ 3 ξ cos 2φ. (28) 2.3. Unirefringent propagation along the racemic ais For û =ẑ, the dadic L(ik û) has the diagonal components [L] 11 = k2 ω 2 (ɛ 1 µ 1 + ξ 2 ) µ 1, (29) [L] 22 = [L] 11, (30) [L] 33 = ω 2 ɛ 3, (31) while [L] lm = 0 for l m. Onl two k-roots emerge from the dispersion relation (21); thus, k 1 = k 3 = ω ɛ 1 µ 1 + ξ 2. (32) Combining the wavenumbers (32) and L(ik ẑ) components (29) (31) with the equation L(ik ẑ) E 0 = 0, we see that the ẑ component of the electric field phasor is null-valued; i.e., ẑ E 0 = E 0 = 0 and E 0 = E 0 ˆ + E 0 ŷ. The time-averaged Ponting vector (9) isgivenb { P(r) t = ep( 2k I) k ( E E 0 2 ) ωξ (E 0 E0 Re + E 0 E } 0) ẑ; (33) 2ω hence k R ẑ P(r) t k=k1 = ω ep( 2k I) 2 [ { ( E E 0 2 ( } ɛ 1 µ 1 + ξ )Re 2 ) k R ẑ P(r) t k=k2 = ω ep ( 2k I) 2 { } ] ξ (E 0 E 0 + E 0 E 0)Re Re{ ɛ µ 1 + ξ 2 }, (34) 1 [ { ( E E 0 2 ( } ɛ 1 µ 1 + ξ )Re 2 ) { } ] ξ +(E 0 E 0 + E 0 E 0)Re Re{ ɛ µ 1 + ξ 2 }. (35) 1 New Journal of Phsics 7 (2005) 165 (

7 7 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Real k a Real k b Imag k a Imag k b (a) (b) Figure 1. Maps of the wavenumbers (normalied with respect to k 0 ) projected onto the θ, φ (0, π/2) surface of the unit sphere. Constitutive parameter values: ɛ 1 = (4.8+i)ɛ 0, ɛ 3 = (1.15+i0.5)ɛ 0, ξ = (5+i4.55) ɛ 0 µ 0, µ 1 = (1.35+i0.7)µ 0 and µ 3 = (3.2+i2)µ 0. (a) The real (top) and imaginar (bottom) parts of the wavenumber k a /k 0. (b) The real (top) and imaginar (bottom) parts of the wavenumber k b /k 0. We therefore find that NPV propagation arises provided that { ( γ 2 ( } { } ɛ 1 µ 1 + ξ +1) Re 2 ) ξ <(γ+ γ ) Re Re { ɛ 1 µ 1 + ξ 2 } for k = k 1, (36) { ( γ 2 ( } ɛ 1 µ 1 + ξ +1)Re 2 ) wherein the comple-valued constant { } ξ < (γ + γ )Re Re{ ɛ µ 1 + ξ 2 } for k = k 2, (37) 1 γ = E 0 E 0 has been introduced for the sake of convenience. (38) New Journal of Phsics 7 (2005) 165 (

8 8 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT no NPV NPV k a Figure 2. Regions of NPV propagation, corresponding to the wavevectors of figure 1, mapped onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 1. NPV propagation for the wavenumber k a is mapped in green. It is instructive to consider the idealied case in which dissipation is neglected. Then, (34) and (35) reduce to kẑ P(r) k=k1 t = kẑ P(r) k=k2 t = ω(ɛ 1µ 1 + ξ 2 )( γ 2 +1) E 0 2. (39) 2µ 1 For propagating planewaves, the inequalit ɛ 1 µ 1 + ξ 2 > 0 (40) must be satisfied. Therefore, we infer that unirefringent NPV propagation along the racemic ais is onl possible in a nondissipative SMCRC material if both ɛ 1 < 0 and µ 1 < 0. Let us note that a similar situation occurs for nondissipative isotropic dielectric-magnetic mediums: therein NPV propagation develops as a result of the permittivit and permeabilit being simultaneousl negative-valued [5] Birefringent propagation Use of the components of L(ik û) givenin(12) (20) along with the time-averaged Ponting vector (9) ields k R û P(r) t = k ( R ep( 2k I û r) 1 2 ω {( E E 0 2 ) cos 2 θ + E 0 2 sin 2 θ { } k [(E 0 E 0 + E 0 E 0 ) cos φ + (E 0 E 0 + E 0 E 0 ) sin φ] sin θ cos θ} Re + 1 { } k ω (E 0 sin φ E 0 cos φ)(e 0 sin φ E 0 cos φ) sin2 θ Re New Journal of Phsics 7 (2005) 165 ( µ 3

9 9 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Real k a 0.6 Real k b Imag k a Imag k b (a) (b) Figure 3. Maps of the wavenumbers (normalied with respect to k 0 ) projected onto the θ, φ (0, π/2) surface of the unit sphere. Constitutive parameter values are as in figure 1, ecept that ξ = (0.3+i9.1) ɛ 0 µ 0. (a) The real (top) and imaginar (bottom) parts of the wavenumber k a /k 0. (b) The real (top) and imaginar (bottom) parts of the wavenumber k b /k 0. + sin θ Re {(E 0 E 0 sin φ + E 0 E 0 cos φ) ξ { }) ξ (E 0 E 0 + E 0 E 0) cos θ Re. (41) After introducing the comple-valued parameters } α = E 0 E 0, β = E 0 E 0, (42) (43) New Journal of Phsics 7 (2005) 165 (

10 10 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT no NPV NPV k a Figure 4. Regions of NPV propagation, corresponding to the wavevectors of figure 3, mapped onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 3. NPV propagation for the wavenumber k a is mapped in green. the epression (41) ma be written as k R û P(r) t = k R ep( 2k I û r) 2 ( 1 ω {( α 2 + β 2 ) cos 2 θ + sin 2 θ [(α + α ) cos φ { k + (β + β ) sin φ] sin θ cos θ} Re { k (α sin φ β cos φ) sin 2 θ Re } µ (α sin φ β cos φ) ω } } + sin θ Re {(α sin φ + β cos φ) ξ { }) ξ (αβ + α β) cos θ Re E µ 0 2. (44) 1 Hence, we have the following condition for birefringent NPV propagation: ( { } 1 k k R ω {( α 2 + β 2 ) cos 2 θ + sin 2 θ [(α + α ) cos φ + (β + β ) sin φ] sin θ cos θ}re + 1 { k ω (α sin φ β cos φ)(α sin φ β cos φ) sin 2 θ Re } { }) + sin θ Re {(α sin φ + β cos φ) ξ ξ (αβ + α β) cos θ Re < 0. µ 3 } (45) New Journal of Phsics 7 (2005) 165 (

11 11 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Real k a Real k b Imag ka Imag k b (a) (b) Figure 5. Maps of the wavenumbers (normalied with respect to k 0 ) projected onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 1, ecept that ɛ 1 = ( 4.8+i)ɛ 0 and ɛ 3 = ( 1.15+i0.5)ɛ 0. (a) The real (top) and imaginar (bottom) parts of the wavenumber k a /k 0. (b) The real (top) and imaginar (bottom) parts of the wavenumber k b /k 0. The quantities α and β ma be derived from the dadic L(ikû) as α = [L] 12[L] 23 [L] 13 [L] 22 [L] 11 [L] 22 [L] 12 [L] 21, (46) β = [L] 23[L] 11 [L] 13 [L] 21 [L] 12 [L] 21 [L] 11 [L] 22. (47) New Journal of Phsics 7 (2005) 165 (

12 12 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT NPV k a,b NPV k a no NPV Figure 6. Regions of NPV propagation, corresponding to the wavevectors of figure 5, mapped onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 5. NPV propagation for the wavenumber k a is mapped in green, whereas NPV propagation for both wavenumbers k a and k b is mapped in red. 3. Numerical results We eplore the NPV inequalit (45) b means of numerical eamples. For illustrative results, let us consider the material specified b the constitutive parameter values ɛ 1 = (4.8+i)ɛ 0, ɛ 3 = (1.15+i0.5)ɛ 0, ξ = (5+i4.55) ɛ 0 µ 0, µ 1 = (1.35+i0.7)µ 0, µ 3 = (3.2+i2)µ 0. (48) While these constitutive parameters are representative of a generic SMCRC material, we stress that the possibilities of fabricating artificial SMCRC materials offer the potential to realie a wide range of constitutive parameter values. The relativel large magnitudes of the magnetoelectric parameters compared with the magnitudes of the permittivit and permeabilit parameters are not uncommon within the realm of NPV propagation: achiral materials with ero permittivit and permeabilit [23, 24] have been generalied to isotropic chiral materials with high magnetoelectric parameters in relation to the permittivit and permeabilit [10, 11, 25]. In figure 1, the values of the two independent wavenumbers k a and k b, as computed using (22) (27), are mapped for propagation directions corresponding to 0 <θ<π/2 and 0 <φ<π/2.we note that the distribution of wavenumbers is smmetric with respect to the plane φ = π/4. The propagation directions which support NPV propagation are presented in figure 2. The NPV propagation directions are clearl confined to the equatorial regions. Furthermore, for this particular eample, NPV onl occurs for the planewave propagation mode specified b the wavenumber k a. Further numerical investigations revealed that reducing the value of ξ, relative to the magnitudes of ɛ 1,3 and µ 1,3, results in NPV propagation being impossible for all propagation New Journal of Phsics 7 (2005) 165 (

13 13 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Real k a Real k b Imag k a Imag k b (a) (b) Figure 7. Maps of the wavenumbers (normalied with respect to k 0 ) projected onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 1, ecept that ɛ 1 = ( 4.8+i)ɛ 0, ɛ 3 = ( 1.15+i0.5)ɛ 0, µ 1 = ( 1.35+i0.7)µ 0 and µ 3 = ( 3.2+i2)µ 0. (a) The real (top) and imaginar (bottom) parts of the wavenumber k a /k 0. (b) The real (top) and imaginar (bottom) parts of the wavenumber k b /k 0. directions. The effect of increasing ξ relative to the magnitudes of ɛ 1,3 and µ 1,3 is illustrated in figures 3 and 4, in which the maps of figures 1 and 2 are repeated for ξ = (0.3+i9.1) ɛ 0 µ 0. While the wavenumbers displaed in figure 3 are substantiall different from those represented in figure 1, the NPV propagation directions portraed in figure 4 are little different from the NPV directions of figure 2. Hence, although the onset of NPV propagation depends criticall upon the relative magnitude of ξ, the NPV propagation directions are little affected b the relative magnitude of ξ. We note that NPV arises onl for the k a planewave mode in figure 4, asisthe case in figure 2. The parameter regimes eplored in figures 1 4 reflect the scenarios which arise in naturalloccurring SMCRC materials insofar as Re {ɛ 1,3 } > 0 and Re {µ 1,3 } > 0. With an ee to the possibilit of fabricating artificial SMCRC materials, we now turn to more general parameter New Journal of Phsics 7 (2005) 165 (

14 14 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT NPV k a,b NPV k b Figure 8. Regions of NPV propagation, corresponding to the wavevectors of figure 7, mapped onto the θ, φ (0,π/2) surface of the unit sphere. Constitutive parameter values are as in figure 7. NPV propagation for the wavenumber k b is mapped in green, whereas NPV propagation for both wavenumbers k a and k b is mapped in red. regimes. In figure 5, the wavenumbers corresponding to the constitutive parameters given in (48), but with ɛ 1 = ( 4.8+i)ɛ 0 and ɛ 3 = ( 1.15+i0.5)ɛ 0 are presented. The distribution of wavenumbers in figure 5 is quite different from that in figures 1 and 3, albeit the smmetr plane at φ = π/4 is a common feature. The corresponding distribution of NPV planewave modes is displaed in figure 6. We see that NPV propagation for the k a mode occurs in both polar regions and in equatorial regions along the general directions of the and aes. Furthermore, there is a relativel large non-polar region, that includes equatorial regions along the general directions of the and aes, for which NPV is supported for both k a and k b modes. Finall, we repeated the calculations of figures 5 and 6, but with µ 1 = ( 1.35+i0.7) µ 0 and µ 3 = ( 3.2+i2) µ 0. The corresponding wavenumber distributions and NPV distributions are given in figures 7 and 8. Clearl, NPV propagation is now widespread, with both k a and k b modes supporting NPV propagation for generall polar directions. Indeed, NPV propagation is supported for all propagation directions. 4. Concluding remarks NPV conditions for both unirefringent and birefringent planewave propagation have been established for materials which ehibit SMCRC behaviour. For scenarios wherein Re{ɛ 1,3 } > 0 and Re{µ 1,3 } > 0 which correspond to the constitutive parameter regimes observed in naturall occurring SMCRC minerals we observe that NPV propagation arises onl if ξ is sufficientl large compared to ɛ 1,3 and µ 1,3. These findings are consistent with those reported for NPV studies in isotropic chiral mediums [10] and Farada chiral mediums [8]. B allowing for the possibilit of Re{ɛ 1,3 } < 0 and Re{µ 1,3 } < 0, we find that occurrence of NPV propagation in terms of propagation modes and their directions becomes much more widespread. Thus, the potential for NPV propagation in artificiall fabricated SMCRC materials is emphasied. New Journal of Phsics 7 (2005) 165 (

15 15 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Acknowledgment We thank an anonmous referee for drawing our attention to a correction in our calculations reported in section 3. References [1] Lakhtakia A, McCall M W and Weiglhofer W S 2003 Negative phase-velocit mediums Introduction to Comple Mediums for Optics and Electromagnetics ed W S Weiglhofer and A Lakhtakia (Bellingham, WA: SPIE Press) pp [2] Shelb R A, Smith D R and Schult S 2001 Eperimental verification of a negative inde of refraction Science [3] Grbic A and Eleftheriades G V 2002 Eperimental verification of backward-wave radiation from a negative inde metamaterial J. Appl. Phs [4] Houck A A, Brock J B and Chuang I L 2003 Eperimental observations of a left-handed material that obes Snell s law Phs. Rev. Lett [5] Pendr J B 2004 Negative refraction Contemp. Phs [6] Hu L and Lin Z 2003 Imaging properties of uniaiall anisotropic negative refractive inde materials Phs. Lett. A [7] Kärkkäinen M K 2003 Numerical stud of wave propagation in uniaiall anisotropic Lorentian backwardwave slabs Phs. Rev. E [8] Macka T G and Lakhtakia A 2004 Plane waves with negative phase velocit in Farada chiral mediums Phs. Rev. E [9] Lakhtakia A 2002 Reversed circular dichroism of isotropic chiral mediums with negative real permeabilit and permittivit Microwave Opt. Technol. Lett [10] Macka T G 2005 Plane waves with negative phase velocit in isotropic mediums Microwave Opt. Technol. Lett ; a corrected version is available at [11] Pendr J B 2004 A chiral route to negative refraction Science [12] Lakhtakia A 1994 Beltrami Fields in Chiral Media (Singapore: World Scientific) [13] Ramakrishna S A 2005 Phsics of negative refractive inde materials Rep. Prog. Phs [14] Lakhtakia A (ed) 1990 Selected Papers on Natural Optical Activit (Bellingham, WA: SPIE Optical Engineering Press) [15] Ro R 1991 Determination of the electromagnetic properties of chiral composites, using normal incidence measurements PhD Thesis Pennslvania State Universit, Universit Park, PA [16] Bose J C 1898 On the rotation of plane of polariation of electric waves b a twisted structure Proc. R. Soc. Lond [17] Whites K W and Chung C Y 1997 Composite uniaial bianisotropic chiral materials characteriation: comparison of prediction and measured scattering J. Electromag. Waves Appl [18] Lakhtakia A and Weiglhofer W S 2000 Electromagnetic waves in a material with simultaneous mirrorconjugated and racemic chiralit characteristics Electromagnetics (The first negative sign on the right side of equation (26) of this paper should be replaced b a positive sign.) [19] Sochava A A, Simovski C R and Tretakov S A 1997 Chiral effects and eigenwaves in bi anisotropic omega structures Advances in Comple Electromagnetic Materials ed A Priou et al (Dordrecht: Kluwer) [20] Gorodetsk G, Hornreich R M and Wankln B M 1973 Statistical mechanics and critical behavior of the magnetoelectric effect in GdVO 4 Phs. Rev. B [21] CookeA H, Swithenb S J and Wells M R 1973 Magnetoelectric measurements on holmium phosphate, HoPO 4 Int. J. Magn New Journal of Phsics 7 (2005) 165 (

16 16 Institute of Phsics DEUTSCHE PHYSIKALISCHE GESELLSCHAFT [22] Lakhtakia A, Varadan V K and Varadan V V 1991 Plane waves and canonical sources in a groelectromagnetic uniaial medium Int. J. Electron [23] Lakhtakia A 2002 An electromagnetic trinit from negative permittivit and negative permeabilit Int. J. Infrared Millim. Waves See also: Lakhtakia A 2002 Int. J. Infrared Millim. Waves [24] Garcia N, Poniovskaa E V and Xiao J Q 2002 Zero permittivit materials: band gaps at the visible Appl. Phs. Lett [25] Tretakov S, Nefedov I, Sihvola A, Maslovski S and Simovski C 2003 Waves and energ in chiral nihilit J. Electromag. Waves Appl New Journal of Phsics 7 (2005) 165 (